On (𝑁,𝑝,𝑞)(𝐸,1) Summability of Fourier Series

Abstract

A new theorem on (𝑁,𝑝,𝑞)(𝐸,1) summability of Fourier series has been established.

1. Introduction

Let {𝑝_𝑛} and {𝑞_𝑛} be the sequences of constants, real or complex, such that

𝑃_𝑛=𝑝₀+𝑝₁+𝑝₂+⋯+𝑝_𝑛=^𝑛_𝜈=0𝑝_𝜈→∞,as𝑄𝑛→∞,_𝑛=𝑞₀+𝑞₁+𝑞₂+⋯+𝑞_𝑛=^𝑛_𝜈=0𝑞_𝜈→∞,as𝑅𝑛→∞,_𝑛=𝑝₀𝑞_𝑛+𝑝₁𝑞_𝑛−1+⋯+𝑝_𝑛𝑞₀=^𝑛_𝜈=0𝑝_𝜈𝑞_𝑛−𝜈→∞,as𝑛→∞. (1.1) Given two sequences{𝑝_𝑛} and {𝑞_𝑛} convolution (𝑝∗𝑞) is defined as 𝑅_𝑛=(𝑝∗𝑞)_𝑛=^𝑛_𝑘=0𝑝_𝑛−𝑘𝑞_𝑘. (1.2) Let ∑^∞_𝑛=0𝑢_𝑛 be an infinite series with the sequence of its 𝑛th partial sums {𝑠_𝑛}.

We write 𝑡_𝑛^𝑝,𝑞=1𝑅_𝑛^𝑛_𝑘=0𝑝_𝑛−𝑘𝑞_𝑘𝑠_𝑘. (1.3)

If 𝑅_𝑛≠0, for all 𝑛, the generalized Nörlund transform of the sequence {𝑠_𝑛} is the sequence {𝑡_𝑛^𝑝,𝑞}.

If 𝑡_𝑛^𝑝,𝑞→𝑆,as𝑛→∞, then the series ∑^∞_𝑛=0𝑢_𝑛 or sequence {𝑠_𝑛} is summable to 𝑆 by generalized Nörlund method (Borwein [1]) and is denoted by

𝑆_𝑛→S(𝑁,𝑝,𝑞). (1.4)

The necessary and sufficient conditions for (𝑁,𝑝,𝑞) method to be regular are ^𝑛_𝑘=0|𝑝_𝑛−𝑘𝑞_𝑘|=𝑂(|𝑅_𝑛|), (1.5)

and 𝑝_𝑛−𝑘=𝑜(|𝑅_𝑛|),as𝑛→∞ for every fixed 𝑘≥0, for which 𝑞_𝑘≠0.

Now 𝐸¹_𝑛=12^𝑛𝑛_𝑘=0𝑛𝑘𝑠_𝑘. (1.6)

If 𝐸¹_𝑛→𝑠,as𝑛→∞, then the series ∑^∞_𝑛=0𝑢_𝑛 is said to be (𝐸,1) summable to 𝑠 (Hardy [2]): 𝑡_𝑛^{𝑝,𝑞,𝐸}=1𝑅_𝑛^𝑛_𝑘=0𝑝_𝑛−𝑘𝑞_𝑘𝐸¹_𝑘=1𝑅_𝑛^𝑛_𝑘=0𝑝_𝑛−𝑘𝑞_𝑘⋅12^𝑘𝑘_𝜈=0𝑘𝜈𝑠_𝜈. (1.7)

If 𝑡_𝑛^{𝑝,𝑞,𝐸}→∞,as𝑛→∞, then we say that the series ∑^∞_𝑛=0𝑢_𝑛 or the sequence {𝑠_𝑛} is summable to 𝑆 by (𝑁,𝑝,𝑞)(𝐸,1) summability method.

Particular Cases
(1)(𝑁,𝑝,𝑞)(𝐸,1) mean reduces to (𝑁,𝑝_𝑛)(𝐸,1) summability mean if 𝑞_𝑛=1,∀𝑛.(2)(𝑁,𝑝,𝑞)(𝐸,1) mean reduces to (𝑁,1/(𝑛+1))(𝐸,1) mean if 𝑝_𝑛=1/(𝑛+1)and𝑞_𝑛=1,∀𝑛.(3)(𝑁,𝑝,𝑞)(𝐸,1) method reduces to (𝑁,𝑞_𝑛)(𝐸,1) if 𝑝_𝑛=1,∀𝑛.(4)(𝑁,𝑝,𝑞)(𝐸,1) method reduces to(𝐶,𝛼)(𝐸,1) if 𝑝_𝑛=(^{𝑛+𝛼−1}_𝛼−1),𝛼>0,and𝑞_𝑛=1,∀𝑛.Let 𝑓(𝑡) be a periodic function with period 2𝜋 and integrable in the sense of Lebesgue over the interval (−𝜋,𝜋).
Let its Fourier series be given by
1𝑓(𝑡)∼2𝑎₀+^∞_𝑛=1(𝑎_𝑛cos𝑛𝑡+𝑏_𝑛sin𝑛𝑡). (1.8) We use the following notations: 𝜙(𝑡)=𝑓(𝑥+𝑡)−𝑓(𝑥−𝑡)−2𝑓(𝑥),Φ(𝑡)=^𝑡₀1|𝜙(𝑢)|𝑑𝑢,𝜏=[𝑡]=theintegralpartof1𝑡,1𝑅(𝑡)=𝑅_𝜏,𝑅_𝑛𝐾=𝑅(𝑛),_𝑛1(𝑡)=2𝜋𝑅_𝑛^𝑛_𝑘=0𝑝_𝑛−𝑘𝑞_𝑘cos^𝑘(𝑡/2)cos(𝑘+1)(𝑡/2)sin.(𝑡/2) (1.9)

2. Theorem

A quite good amount of work is known for Fourier series by ordinary summability method. The purpose of this paper is to study Fourier series by (𝑁,𝑝,𝑞)(𝐸,1) summability method in the following form.

Theorem 2.1. Let {𝑝_𝑛}and{𝑞_𝑛} be positive monotonic, nonincreasing sequences of real numbers such that 𝑅_𝑛=^𝑛_𝑘=0𝑝_𝑘𝑞_𝑛−𝑘→∞,as𝑛→∞. (2.1) Let 𝛼(𝑡) be a positive, nondecreasing function of 𝑡. If Φ(𝑡)=^𝑡₀𝑡|𝜙(𝑢)|𝑑𝑢=𝑜(𝛼(1/𝑡)),as𝑡→+0,𝛼(𝑛)→∞,as𝑛→∞, (2.2)(2.3) then a sufficient condition that the Fourier Series (1.8) be summable (𝑁,𝑝,𝑞)(𝐸,1) to 𝑓(𝑥) at the point 𝑡=𝑥 is ^𝑛₁𝑅(𝑢)𝑢𝛼(𝑢)𝑑𝑢=𝑂(𝑅_𝑛),as𝑛→∞. (2.4)

3. Lemmas

Proof of the theorem needs some lemmas.

Lemma 3.1. For 0≤𝑡≤1/𝑛,|𝐾_𝑛(𝑡)|=𝑂(𝑛). (3.1)

Proof. |𝐾_𝑛1(𝑡)|=2𝜋𝑅_𝑛|^𝑛_𝑘=0𝑝_𝑛−𝑘𝑞_𝑘cos^𝑘(𝑡/2)sin(𝑘+1)(𝑡/2)sin|≤1(𝑡/2)2𝜋𝑅_𝑛^𝑛_𝑘=0𝑝_𝑛−𝑘𝑞_𝑘(𝑘+1)|sin(𝑡/2)||sin1(𝑡/2)|=𝑂(𝑛)𝑅_𝑛^𝑛_𝑘=0𝑝_𝑛−𝑘𝑞_𝑘=𝑂(𝑛). (3.2)

Lemma 3.2. If {𝑝_𝑛} and {𝑞_𝑛} are nonnegative and nonincreasing, then for 0≤𝑎≤𝑏<∞, 0≤𝑡≤𝜋, and any 𝑛 we have 12𝜋𝑅_𝑛|^𝑏_𝑘=𝑎𝑝_𝑛−𝑘𝑞_𝑘cos^𝑘(𝑡/2)sin(𝑘+1)(𝑡/2)𝑅sin(𝑡/2)|=𝑂(^𝜏𝑡𝑅_𝑛). (3.3)

Proof. 12𝜋𝑅_𝑛|^𝑏_𝑘=𝑎𝑝_𝑛−𝑘𝑞_𝑘cos^𝑘(𝑡/2)sin(𝑘+1)(𝑡/2)sin|≤1(𝑡/2)𝑡𝜋𝑅_𝑛|^𝑏_𝑘=𝑎𝑝_𝑛−𝑘𝑞_𝑘cos^𝑘(𝑡2)sin𝑡(𝑘+1)2|=1𝑡𝜋𝑅_𝑛|Im{^𝑏_𝑘=𝑎𝑝_𝑛−𝑘𝑞_𝑘cos^𝑘(𝑡2)𝑒^{𝑖(𝑘+1)(𝑡/2)}≤1}|𝑡𝜋𝑅_𝑛|^𝑏_𝑘=𝑎𝑝_𝑛−𝑘𝑞_𝑘cos^𝑘(𝑡2)𝑒^{𝑖𝑘𝑡/2}||𝑒^𝑖𝑡/2|≤1𝑡𝜋𝑅_𝑛|^𝑏_𝑘=𝑎𝑝_𝑛−𝑘𝑞_𝑘cos^𝑘(𝑡2)𝑒^{𝑖𝑘𝑡/2}|≤1𝑡𝜋𝑅_𝑛{|^𝜏−1_𝑘=𝑎𝑝_𝑛−𝑘𝑞_𝑘cos^𝑘(𝑡2)𝑒𝑖𝑘𝑡2|+|^𝑏_𝑘=𝜏𝑝_𝑛−𝑘𝑞_𝑘cos^𝑘(𝑡2)𝑒^𝑖𝑘(𝑡2⁾|}. (3.4) Now considering first term of (3.4), we have 1𝑡𝜋𝑅_𝑛|^𝜏−1_𝑘=𝑎𝑝_𝑛−𝑘𝑞_𝑘cos^𝑘(𝑡2)𝑒^{𝑖𝑘(𝑡/2)}1|≤𝑡𝜋𝑅_𝑛^𝜏−1_𝑘=𝑎𝑝_𝑛−𝑘𝑞_𝑘|𝑒^{𝑖𝑘(𝑡/2)}1|≤𝑡𝜋𝑅_𝑛^𝜏−1_𝑘=𝑎𝑝_𝑛−𝑘𝑞_𝑘≤1𝑡𝜋𝑅_𝑛^𝜏−1_𝑘=𝑎𝑝_𝜏−𝑘𝑞_𝑘≤1𝑡𝜋𝑅_𝑛(𝑅_𝜏𝑅)=𝑂(^𝜏𝑡𝑅_𝑛). (3.5) Now considering second term of (3.4) and using Abel’s lemma, we have 1𝑡𝜋𝑅_𝑛|^𝑏_𝑘=𝜏𝑝_𝑛−𝑘𝑞_𝑘cos^𝑘(𝑡2)𝑒^{𝑖𝑘(𝑡/2)}1|≤𝑡𝜋𝑅_𝑛|^𝑏_𝑘=𝜏𝑝_𝑛−𝑘𝑞_𝑘𝑒^{𝑖𝑘(𝑡/2)}|≤2𝑝_𝑛−𝑏𝑞_𝜏𝑡𝜋𝑅_𝑛max_{𝜏+1≤𝑘≤𝑏}|1−𝑒^{𝑖(𝑘+1)(𝑡/2)}1−𝑒^𝑖𝑡/2|≤4𝑝_𝑛−𝑏𝑞_𝜏𝑡𝜋𝑅_𝑛|𝑒^{−𝑖𝑡/4}𝑒^𝑖𝑡/4−𝑒^{−𝑖𝑡/4}|≤2𝑞_𝜏𝑡𝜋𝑅_𝑛(𝑝_𝑛−𝑏𝑃_𝜏)𝑃_𝜏|1sin(𝑡/4)|(where𝑃_𝜏=^𝜏_𝑘=0𝑝_𝜏−𝑘)≤8𝑞_𝜏𝑡𝜋𝑅_𝑛(𝑝_𝑛−𝑏𝑃_𝜏)𝑃_𝜏|1𝑡|≤8𝑞_𝜏𝑃_𝜏𝑡𝜋𝑅_𝑛≤8𝑅_𝜏𝑡𝜋𝑅_𝑛(since𝑅_𝜏=^𝜏_𝑘=0𝑝_𝜏−𝑘𝑞_𝑘≥𝑃_𝜏𝑞_𝜏)𝑅=𝑂(^𝜏𝑡𝑅_𝑛). (3.6) Using (3.5) and (3.6), we get the required result of Lemma 3.2.

4. Proof of Theorem

Following Zygmund [3], the 𝑛th partial sum 𝑠_𝑛(𝑥) of the series (1.8) at 𝑡=𝑥 is given by 𝑠_𝑛1(𝑥)=𝑓(𝑥)+2𝜋^𝜋₀𝜙_𝑥(𝑡)sin(𝑛+1/2)𝑡sin(𝑡/2)𝑑𝑡. (4.1)

So the (𝐸,1) mean of the series (1.8) at𝑡=𝑥 is given by 𝐸¹_𝑛1(𝑥)=2^𝑛𝑛_𝑘=0𝑛𝑘𝑠_𝑘1(𝑥)=𝑓(𝑥)+2^𝑛+1𝜋^𝜋₀𝜙_𝑥(𝑡)sin{(𝑡/2)^𝑛_𝑘=0𝑛𝑘sin1(𝑘+21)𝑡}𝑑𝑡=𝑓(𝑥)+2^𝑛+1𝜋^𝜋₀𝜙_𝑥(𝑡)sin(𝑡/2)Im{𝑒^𝑖𝑡/2(1+𝑒^𝑖𝑡)^𝑛1}𝑑𝑡=𝑓(𝑥)+2^𝑛+1𝜋^𝜋₀𝜙_𝑥(𝑡)sin(𝑡/2)Im{𝑒^𝑖𝑡/2(1+cos𝑡+𝑖sin𝑡)^𝑛1}𝑑𝑡=𝑓(𝑥)+2^𝑛+1𝜋^𝜋₀𝜙_𝑥(𝑡)sin(𝑡/2)Im{𝑒^𝑖𝑡/22^𝑛cos^𝑛(𝑡2𝑡)(cos2+𝑖sin𝑡2)^𝑛1}𝑑𝑡=𝑓(𝑥)+2^𝑛+1𝜋^𝜋₀𝜙_𝑥(𝑡)sin(𝑡/2)Im{𝑒^𝑖𝑡/22^𝑛cos^𝑛(𝑡2){(cos𝑛𝑡2+𝑖sin𝑛𝑡21)}}𝑑𝑡=𝑓(𝑥)+2𝜋^𝜋₀𝜙_𝑥(𝑡)cos^𝑛(𝑡/2)sin(𝑛+1)(𝑡/2)sin(𝑡/2)𝑑𝑡. (4.2)

Therefore (𝑁,𝑝,𝑞) transform of {𝐸¹_𝑛(𝑥)} is given by 𝑡_𝑛^{𝑝,𝑞,𝐸}1(𝑥)=𝑅_𝑛^𝑛_𝑘=0𝑝_𝑛−𝑘𝑞_𝑘𝐸¹_𝑘1(𝑥)=𝑓(𝑥)+2𝜋^𝜋₀1𝑅_𝑛^𝑛_𝑘=0𝑝_𝑛−𝑘𝑞_𝑘𝜙_𝑥(𝑡)cos^𝑘(𝑡)sin(𝑘+1)(𝑡/2)sin(𝑡/2)=𝑓(𝑥)+^𝜋₀𝐾_𝑛(𝑡)𝜙_𝑥𝑡(𝑡)𝑑𝑡,_{𝑛𝑝,𝑞,𝐸}(𝑥)−𝑓(𝑥)=[₀^1/𝑛+^𝛿_1/𝑛+^𝜋_𝛿]𝐾_𝑛(𝑡)𝜙_𝑥(𝑡)𝑑𝑡=𝐼₁+𝐼₂+𝐼₃(say). (4.3) We have |𝐼₁|≤₀^1/𝑛|𝐾_𝑛(𝑡)||𝜙_𝑥(𝑡)|𝑑𝑡=𝑂(𝑛)₀^1/𝑛|𝜙_𝑥(𝑡)|𝑑𝑡(usingLemma13.1)=𝑂(𝑛)𝑜(𝑛𝛼(𝑛))(by1(2.2))=𝑜(𝛼(𝑛))=𝑜(1)as𝑛→∞(by(2.3)). (4.4)

Now we consider |𝐼₂|≤^𝛿_1/𝑛|𝐾_𝑛(𝑡)||𝜙_𝑥(𝑡)|𝑑𝑡(where=0<𝛿<1)_𝛿1/𝑛𝑂(𝑅(1/𝑡)𝑡𝑅_𝑛)|𝜙_𝑥(𝑡)|𝑑𝑡(usingLemma13.2)=𝑂(𝑅_𝑛)^𝛿_1/𝑛(𝑅(1/𝑡)𝑡)|𝜙_𝑥1(𝑡)|𝑑𝑡=𝑂(𝑅_𝑛)[{(𝑅(1/𝑡)𝑡)𝜙_𝑥(𝑡)}^𝛿_1/𝑛−^𝛿_1/𝑛𝑑(𝑅(1/𝑡)𝑡)𝜙_𝑥1(𝑡)]=𝑂(𝑅(𝑛))[{𝑜(𝑅(1/𝑡)𝛼(1/𝑡))}^𝛿_1/𝑛−^𝛿_1/𝑛𝜙_𝑥(𝑡)𝑑(𝑅(1/𝑡)𝑡)](by1(2.1))=𝑜(1𝑅(𝑛))+𝑜(1𝛼(𝑛))+𝑜(𝑅(𝑛))[^𝛿_1/𝑛𝜙_𝑥(𝑡){𝑑(𝑅(1/𝑡)𝛼(1/𝑡)1𝑡𝛼(1/𝑡))}]=𝑜(1𝑅(𝑛))+𝑜(1𝛼(𝑛))+𝑜()𝑅(𝑛)×[^𝛿_1/𝑛𝑡𝑜(𝛼(1/𝑡)){𝑑𝛼(1/𝑡)(𝑅(1/𝑡)1𝑡𝛼(1/𝑡))}+𝛼(𝑡)𝑑(𝑅(1/𝑡)1𝑡𝛼(1/𝑡))]=𝑜(1𝑅(𝑛))+𝑜(𝛼(𝑛))+𝑜(1)[^𝛿_1/𝑛𝑑𝛼(1/𝑡){𝛼(1/𝑡)}²1+𝑜()𝑅(𝑛)^𝛿_1/𝑛𝑡𝑑(𝑅(1/𝑡)1𝑡𝛼(1/𝑡))]=𝑜(1𝑅(𝑛))+𝑜(1𝛼(𝑛))+𝑜(1){}𝛼(1/𝑡)^𝛿_1/𝑛1+𝑜(𝑅(𝑛))[{𝑡𝑅(1/𝑡)}𝑡𝛼(1/𝑡)^𝛿_1/𝑛−^𝛿_1/𝑛(𝑅(1/𝑡)1𝑡𝛼(1/𝑡))𝑑𝑡]=𝑜(1𝑅(𝑛))+𝑜(1𝛼(𝑛))+𝑜(1𝛼(𝑛))+𝑜()𝑅(𝑛)¹_1/𝑛(𝑅(1/𝑡)1𝑡𝛼(1/𝑡))𝑑𝑡=𝑜(1𝑅(𝑛))+𝑜(1𝛼(𝑛))+𝑜()𝑅(𝑛)^𝑛₁(𝑅(𝑢)1𝑢𝛼(𝑢))𝑑𝑢{∵𝑡1=𝑢}=𝑜(1𝑅(𝑛))+𝑜(1𝛼(𝑛))+𝑜(𝑅(𝑛))𝑂(𝑅_𝑛)(by1(2.4))=𝑜(1𝑅(𝑛))+𝑜(𝛼(𝑛))+𝑜(1)=𝑜(1),as𝑛→∞(byvirtueof(2.1)and(2.2)). (4.5)

Now by Riemann-Lebesgue theorem and by regularity of the method of summability we have

𝐼₃=^𝜋_𝛿|𝑘_𝑛(𝑡)||𝜙_𝑥(𝑡)|𝑑𝑡=𝑜(1),as𝑛→∞. (4.6)

This completes the proof of the theorem.

5. Corollaries

Following corollaries can be derived from our main theorem.

Corollary 5.1. If 𝑡Φ(𝑡)=𝑜[log(1/𝑡)],as𝑡→+0, (5.1) then the Fourier series (1.8) is (𝐶,1)(𝐸,1) summable to 𝑓(𝑥) at the point 𝑡=𝑥.

Corollary 5.2. If Φ(𝑡)=𝑜(𝑡),as𝑡→+0, (5.2) then the Fourier series (1.8) is (𝑁,𝑝_𝑛)(𝐸,1) summable to 𝑓(𝑥) at the point 𝑡=𝑥, provided that {𝑝_𝑛} be a positive, monotonic, and nonincreasing sequence of real numbers such that 𝑝_𝑛=𝑝₀+𝑝₁+⋯+𝑝_𝑛→∞,as𝑛→∞. (5.3)

Acknowledgments

The authors are grateful to Professor Shyam Lal, Department of Mathematics, Harcourt Butler Technological Institute, Kanpur, India, for his valuable suggestions and guidence in preparation of this paper. The authors are also thankful to Professor M. P. Jain, Vice Chancellor, Mody Institute of Technology and Science (Deemed University), Lakshmangarh, Sikar, Rajasthan, India, and to Professor S. N. Puri, Former Dean, Faculty of Engineering, Technology, Mody Institute of Technology and Science (Deemed University), Laxmangarh, Sikar, Rajasthan.